The last rodeo I shot I used my Sigma 70-300 and due to that I'm now shopping for a 70-200L 2.8. Are you shooting from the stands? Up above in announcer/press box or arena level at the arena fencing?
IMHO unless you're shooting length wise across the arena your 70-200 should be fine. Have you been to rodeo at all or understand the events? The steers are quick and the field of view for your shot is going to be wide once the header and heeler get stretched out.
Here's just a quick one I processed. 81mm focal length. f4 @ 1/160th ISO 1600. f2.8 would have helped me out tons.
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It would help if you knew the arena and setup but I would guess your 70-200 will be just fine. Stuff happens quick. Hope you're better at panning than me

5DIII | 40D | 17-40 f4L | Tamron 28-75 2.8 | 50 1.4 | 70-200 2.8L | Oly Zuiko 50 macro | Tamron 1.4x